Cyclic (r, ) locally recoverable codes from their constacyclic and negacyclic subcodes

If a symbol in any coordinate of a codeword in a code C can be repaired by accessing at most r other coordinates, then the positive integer r is called locality of the code. Codes with locality are called locally recoverable codes. Locally recoverable codes are preferred in distributed storage systems, such as Microsoft Azure and Hadoop (used by Facebook), due to their ability to recover a failed node by accessing the minimum number of surviving nodes. A code with (r, ) -locality is a locally recoverable code that allows recovering -1 erasures simultaneously by reaching at most r other coordinates. In this paper, we obtained constacyclic and negacyclic codes by determining the structure of cyclotomic cosets. Then, we constructed cyclic (r, ) -LRCs by virtue of their constacyclic and negacyclic subcodes which we found.